Estimating a Population Proportion and Confidence Intervals

7.1 Estimating a Population Proportion

Introduction

Estimating a population proportion is a fundamental concept in inferential statistics. It involves using sample data to make inferences about the proportion of a population that possesses a certain characteristic. This section covers point estimation, confidence intervals, margin of error, and sample size determination for population proportions.

Point Estimate

A point estimate is a single value used to estimate a population parameter. For population proportions, the sample proportion, denoted as p̂, is the best point estimate of the true population proportion p.

• Definition: The sample proportion p̂ is calculated as the number of successes divided by the total sample size.

• Formula: $\hat{p} = \frac{x}{n}$ where $x$ is the number of successes and $n$ is the sample size.

• Example: If 329 out of 362 drive-thru orders at McDonald's are accurate, then $\hat{p} = \frac{329}{362} = 0.0912$.

Confidence Interval for a Population Proportion

A confidence interval provides a range of values that is likely to contain the true population proportion. It is more informative than a point estimate because it accounts for sampling variability.

• Definition: A confidence interval is a range (or interval) of values used to estimate the value of a population parameter.

• Notation: Confidence intervals are sometimes denoted by CI.

• Confidence Level: The probability that the confidence interval contains the true population parameter. Common levels are 90%, 95%, and 99%.

• Formula: $\hat{p} - E < p < \hat{p} + E$ where $E$ is the margin of error.

• Example: For a 95% confidence interval, if $\hat{p} = 0.0912$ and $E = 0.0297$, then $0.0615 < p < 0.121$.

Interpreting Confidence Intervals

It is essential to interpret confidence intervals correctly. The confidence level refers to the proportion of intervals that will contain the true population parameter if the sampling process is repeated many times.

• Correct Interpretation: "We are 95% confident that the interval 0.0615 to 0.121 contains the true value of the population proportion."

• Common Mistake: Saying "There is a 95% chance that the true value is between 0.0615 and 0.121" is incorrect. The true value is fixed; the interval may or may not contain it.

• Visual Example: A diagram (Figure 7-1) shows confidence intervals from 20 samples, illustrating that not all intervals contain the true proportion.

Critical Values

Critical values are used to determine the endpoints of confidence intervals. They separate significant values from non-significant ones in the standard normal distribution.

• Definition: A critical value is a value that separates sample statistics from significantly low or high values.

• Notation: $z_{\alpha/2}$ is the critical value for a given confidence level.

• Table of Common Critical Values:

Confidence Level	α	$z_{\alpha/2}$
90%	0.10	1.645
95%	0.05	1.96
99%	0.01	2.575

• Example: For a 90% confidence level, $z_{\alpha/2} = 1.645$.

Margin of Error

The margin of error is the maximum likely difference between the sample proportion and the population proportion.

• Formula: $E = z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

• Example: If $\hat{p} = 0.0912$, $n = 362$, and $z_{\alpha/2} = 1.96$, then $E = 1.96 \sqrt{\frac{0.0912 \times 0.9088}{362}} = 0.0297$

Steps to Constructing a Confidence Interval for a Population Proportion

• 1. Verify requirements: Sample must be simple random, fixed number of trials, independent outcomes, and probabilities are constant.

• 2. Compute the sample proportion $\hat{p}$.

• 3. Find the critical value $z_{\alpha/2}$ for the desired confidence level.

• 4. Calculate the margin of error $E$.

• 5. Construct the confidence interval: $\hat{p} - E < p < \hat{p} + E$.

• 6. Round to three significant digits.

Analyzing Polls and Surveys

Polls are commonly used to estimate population proportions. However, the reliability of polls depends on the sampling method and sample size.

• 1. The sample should be a simple random sample, not a voluntary response sample.

• 2. The confidence level must be stated.

• 3. The quality of poll results depends on the sampling method and sample size.

Determining Sample Size

Calculating the required sample size is important for estimating a population proportion with a specified margin of error.

• Formula (if $\hat{p}$ is known): $n = \frac{\hat{p}(1-\hat{p})z_{\alpha/2}^2}{E^2}$

• Formula (if $\hat{p}$ is unknown): $n = \frac{0.25z_{\alpha/2}^2}{E^2}$

• Rounding: Always round up to the next whole number.

• Example: To estimate the percentage of California residents that are left-handed with a margin of error of 3% and 95% confidence level, $n = \frac{0.25 \times (1.96)^2}{(0.03)^2} = 1067$

Calculator Instructions

Many calculators have built-in functions to compute confidence intervals for proportions. For example, on the TI-83/84 Plus:

• 1. Press STAT, then select TESTS.

• 2. Select 1-PropZInt in the menu.

• 3. Enter the number of successes, number of observations, and confidence level.

• 4. Select Calculate and press Enter.

Additional Examples

• Example: Fermi groups of 2 estimate the sample size needed to estimate the proportion of times a coin comes up heads with 80% confidence and a margin of error of 0.08.

• Application: In a survey of 1002 people, 70% say they voted. Voting records show that 61% of eligible people voted. Constructing a confidence interval for the proportion of people who say they voted yields $0.666 < p < 0.734$. Since 61% is not in the interval, we conclude that more people say they voted than actually did.

Summary Table: Confidence Levels and Critical Values

Confidence Level	α	$z_{\alpha/2}$
90%	0.10	1.645
95%	0.05	1.96
99%	0.01	2.575

Key Points to Remember

• Always verify sample requirements before constructing a confidence interval.

• Use the appropriate critical value for the desired confidence level.

• Round sample size up to the next whole number.

• The sample size does not depend on the population size.

• Interpret confidence intervals correctly: the confidence level refers to the method, not the probability that the interval contains the parameter.

Additional info: The confidence interval described here is known as the Wald confidence interval. There are other methods for constructing confidence intervals for proportions, such as the Wilson score and Clopper-Pearson intervals, which may be more accurate for small samples or extreme proportions.